Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F=(F1,…,Fm){F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹n{\mathbb{B}^{n}}, n≥2{n\geq 2}, to a bounded symmetric domain Ω=Ω1×⋯×Ωm{\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ωi′{\Omega_{i}^{\prime}}s are irreducible factors of Ω. We prove every non-constant component Fi{F_{i}} must map generic boundary points of 𝔹n{\mathbb{B}^{n}} to the boundary of Ωi{\Omega_{i}}. In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.